fluid_mechanics

634 Chapter 11 ■ Compressible Flow
Compressible gas
flows and openchannel
liquid
flows are strikingly
similar in several
ways.
and inexpensively demonstrated with a shallow, open-channel flow field in a ripple tank or water
table.
The propagation of weak pressure pulses 1sound waves2 in a compressible flow can be
considered to be comparable to the movement of small amplitude waves on the surface of an
open-channel flow. In each case—two-dimensional compressible flow and open-channel flow—
the influence of flow velocity on wave pattern is similar. When the flow velocity is less than
the wave speed, wave fronts can move upstream of the wave source and the flow is subsonic
1compressible flow2 or subcritical 1open-channel flow2. When the flow velocity is equal to the
wave speed, wave fronts cannot move upstream of the wave source and the flow is sonic 1compressible
flow2 or critical 1open-channel flow2. When the flow velocity is greater than the wave
speed, the flow is supersonic 1compressible flow2 or supercritical 1open-channel flow2. Normal
shocks can occur in supersonic compressible flows. Hydraulic jumps can occur in supercritical
open-channel flows. Comparison of the characteristics of normal shocks 1Section 11.5.32 and
hydraulic jumps 1Section 10.6.12 suggests a strong resemblance and thus analogy between the
two phenomena.
For compressible flows a meaningful dimensionless variable is the Mach number, where
In open-channel flows, an important dimensionless variable is the Froude number, where
(11.46)
(11.157)
The velocity of the channel flow is V oc , the acceleration of gravity is g, and the depth of the flow
is y. Since the speed of a small amplitude wave on the surface of an open-channel flow, c oc , is 1see
Section 10.2.12
we conclude that
Ma V c
Fr V oc
1gy
c oc 1gy
(11.158)
Fr V oc
c oc
(11.159)
From Eqs. 11.46 and 11.159 we see the similarity between Mach number 1compressible flow2 and
Froude number 1open-channel flow2.
For compressible flow, the continuity equation is
rAV constant
(11.160)
where V is the flow velocity, r is the fluid density, and A is the flow cross-sectional area. For an
open-channel flow, conservation of mass leads to
ybV oc constant
(11.161)
where V oc is the flow velocity, and y and b are the depth and width of the open-channel flow. Comparing
Eqs. 11.160 and 11.161 we note that if flow velocities are considered similar and flow area,
A, and channel width, b, are considered similar, then compressible flow density, r, is analogous to
open-channel flow depth, y.
It should be pointed out that the similarity between Mach number and Froude number is generally
not exact. If compressible flow and open-channel flow velocities are considered to be similar,
then it follows that for Mach number and Froude number similarity the wave speeds c and c oc
must also be similar.
From the development of the equation for the speed of sound in an ideal gas 1see Eqs. 11.34
and 11.352 we have for the compressible flow
c 21constant2 kr k1
(11.162)
From Eqs. 11.162 and 11.158, we see that if y is to be similar to r as suggested by comparing Eq.
11.160 and 11.161, then k should be equal to 2. Typically k 1.4 or 1.67, not 2. This limitation